Hartogs's theorem on separate holomorphicity

Hartogs's theorem states that a function holomorphic in each variable separately on a domain in complex Euclidean space is jointly holomorphic, under standard domain assumptions.

Connect

Cauchy's integral formula: An integral identity expressing a holomorphic function's interior values through its values on a surrounding contour. Iterated Cauchy formulas provide the local analytic structure behind the theorem.
Separate holomorphicity: Holomorphic dependence on each coordinate separately while the other coordinates are held fixed. It is the coordinatewise hypothesis whose strength Hartogs's theorem establishes.
Friedrich Hartogs: A German mathematician whose work helped establish the theory of functions of several complex variables. The theorem bears Hartogs's name and belongs to the research program he advanced.
Separate analyticity: Analytic dependence on each variable individually, without initially requiring joint analyticity. It is the broader coordinatewise condition from which the theorem derives joint holomorphicity.
Local boundedness: The property that a function is bounded on some neighborhood of every point in its domain. Local boundedness is a key route from separate holomorphicity to joint holomorphicity.
Joint holomorphicity: Holomorphic dependence on several complex variables simultaneously. It is the full multivariable conclusion, stronger than separate holomorphicity alone appears to guarantee.
Hartogs's 1906 theorem: Hartogs's result that separately holomorphic functions in several complex variables are jointly holomorphic under suitable assumptions. This historical formulation is the source of the modern separate-holomorphicity theorem.
Separate continuity: Continuity in each variable individually while other variables are fixed. Unlike separate holomorphicity, separate continuity alone does not ensure joint continuity.
Power series: An infinite series of powers of variables that can represent a function within a region of convergence. Joint holomorphicity means the function admits convergent local power-series expansions.
Domain (mathematics): A connected open subset of a mathematical space, especially of complex Euclidean space. The theorem's standard formulation assumes the function is defined on an open domain.